Maxwell Relation, Gibbs Free Energy, Thermal Expansion Coefficient

In summary, by using a Maxwell relation derived from the Gibbs free energy and incorporating the third law of thermodynamics, it can be proven that the thermal expansion coefficient β must be zero at T = 0. This can be shown without relying on the ideal gas law.
  • #1
Tian En

Homework Statement


By means of a Maxwell relation derived from the Gibbs free energy and making use of the third law of thermodynamics, prove that the thermal expansion coefficient β must be zero at T = 0. I tried but I got something funny.

Homework Equations


$$G=U-TS+PV$$
$$dG=\mu dN-SdT+VdP$$
$$S=Nk_B[\ln(\frac{V}{N}(\frac{4\pi mU}{3Nh^2})^{3/2})+\frac{5}{2}]$$
$$PV=Nk_B T$$
$$\beta = \frac{1}{V}\frac{\partial V}{\partial T} \Bigg| _{N,P}$$

The Attempt at a Solution


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  • #2
I believe you have essentially shown that an ideal gas is not compatible with the third law.

I think you should be able to finish the proof using your result
upload_2017-10-26_11-51-36.png
without assuming the ideal gas law.
 

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Last edited:

1. What is the Maxwell Relation and how is it used in thermodynamics?

The Maxwell Relation is a mathematical relationship between thermodynamic properties that allows for the calculation of one property from the derivatives of another. It is used to simplify calculations and determine the relationship between different thermodynamic variables.

2. What is Gibbs Free Energy and why is it important in thermodynamics?

Gibbs Free Energy is a thermodynamic property that measures the amount of energy available in a system to do work. It is important in thermodynamics because it helps predict whether a process will occur spontaneously or not, and at what conditions.

3. How is Gibbs Free Energy related to the concept of spontaneity?

In thermodynamics, a process is considered spontaneous if it occurs without any external intervention. Gibbs Free Energy is related to spontaneity because a negative change in Free Energy (ΔG) indicates that a process is spontaneous and can occur without the input of external energy.

4. What is the thermal expansion coefficient and how does it relate to temperature changes?

The thermal expansion coefficient is the measure of how much a material expands or contracts when its temperature changes. It is directly proportional to temperature changes, meaning that as temperature increases, the material will expand, and as temperature decreases, the material will contract.

5. How is the thermal expansion coefficient used in practical applications?

The thermal expansion coefficient is used in various practical applications, such as in the design of bridges, buildings, and other structures, where temperature changes can cause expansion or contraction. It is also used in the production of materials with specific thermal properties, such as heat-resistant metals for aerospace applications.

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